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June 2009 Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions
Vuk Milišić
Methods Appl. Anal. 16(2): 157-186 (June 2009).


This work is a continuation of [3]; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In [3] we studied a priori estimates in this setting; here we fully develop very weak estimates à la Nečas [17] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [3]. We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [13, 18].


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Vuk Milišić. "Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions." Methods Appl. Anal. 16 (2) 157 - 186, June 2009.


Published: June 2009
First available in Project Euclid: 2 November 2009

zbMATH: 1281.76052
MathSciNet: MR2563746

Primary: 35B27 , 65Mxx , 76D05 , 76Mxx

Keywords: boundary layers , error estimates , Laplace equation , multi-scale modelling , natural boundary conditions , rough boundary , vertical boundary correctors , Wall-laws

Rights: Copyright © 2009 International Press of Boston


Vol.16 • No. 2 • June 2009
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